The Stacks project

Lemma 47.18.4. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring. Let $\omega _ A^\bullet $ be a normalized dualizing complex. Let $E$ be an injective hull of the residue field. Let $K \in D_{\textit{Coh}}(A)$. Then

\[ \mathop{\mathrm{Ext}}\nolimits ^{-i}_ A(K, \omega _ A^\bullet )^\wedge = \mathop{\mathrm{Hom}}\nolimits _ A(H^ i_{\mathfrak m}(K), E) \]

where ${}^\wedge $ denotes $\mathfrak m$-adic completion.

Proof. By Lemma 47.15.3 we see that $R\mathop{\mathrm{Hom}}\nolimits _ A(K, \omega _ A^\bullet )$ is an object of $D_{\textit{Coh}}(A)$. It follows that the cohomology modules of the derived completion of $R\mathop{\mathrm{Hom}}\nolimits _ A(K, \omega _ A^\bullet )$ are equal to the usual completions $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, \omega _ A^\bullet )^\wedge $ by More on Algebra, Lemma 15.94.4. On the other hand, we have $R\Gamma _{\mathfrak m} = R\Gamma _ Z$ for $Z = V(\mathfrak m)$ by Lemma 47.10.1. Moreover, the functor $\mathop{\mathrm{Hom}}\nolimits _ A(-, E)$ is exact hence factors through cohomology. Hence the lemma is consequence of Theorem 47.18.3. $\square$

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